Bounds for the class of nilpotent wreath products

1966 ◽  
Vol 62 (2) ◽  
pp. 165-169 ◽  
Author(s):  
Teresa Scruton
Keyword(s):  
Wreath Product ◽  
Direct Product ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Cyclic Groups ◽  
Image Position ◽  
Finite Exponent

Introduction. In his paper ((1)), Baumslag has shown that the wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group, the prime p being the same for both groups. Liebeck ((3)) has obtained the exact nilpotency class of A wr B when A and B are Abelian. Let A be an Abelian p -group of exponent pn and let B be a direct product of cyclic groups, whose orders are pβ1, …, pβn, with β1 ≤ β2 ≤ … βn. Then A wr B has nilpotency class .

On nilpotent wreath products

1970 ◽  
Vol 68 (1) ◽  
pp. 1-15 ◽  
Author(s):  
J. D. P. Meldrum
Keyword(s):  
Wreath Product ◽  
Direct Product ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Cyclic Groups ◽  
Image Position ◽  
Finite Exponent

1. Introduction. The wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p (Baumslag (1)). When A is an Abelian p-group of exponent pk, and B is the direct product of cyclic groups of orders pβ1, …, pβn and β1 ≥ β2 ≥ …, ≥ βn, then Liebeck has shown that the nilpotency class c of A wr B is

Central series in wreath products

1967 ◽  
Vol 63 (3) ◽  
pp. 551-567 ◽  
Author(s):  
J. D. P. Meldrum
Keyword(s):  
Wreath Product ◽  
Abelian Groups ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Central Series ◽  
Base Group ◽  
Finite Exponent

In this paper we study the structure of the α-central series of the nilpotent wreath product of two Abelian groups, the α-central series being the intersection of the upper central series with the base group. Let C = A wr B, the standard restricted wreath product of A and B. Then Baumslag(1) showed that C is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p. In (5) Liebeck calculated the nilpotency class of the nilpotent wreath product of two Abelian groups. We obtain an expression for an element of the base group to belong to a given term of the upper central series.

Concerning nilpotent wreath products

1962 ◽  
Vol 58 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Hans Liebeck
Keyword(s):  
Lower Bound ◽  
Wreath Product ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Nilpotency Class ◽  
Simple Construction ◽  
Metabelian Groups ◽  
Engel Condition ◽  
Finite Exponent

In a recent paper, (2), Baumslag showed that a wreath product of A by B is nilpotent if and only if A and B are nilpotent p-groups for the same prime p, with A of finite exponent and B finite. We shall calculate the (nilpotency) class of such groups when A and B are Abelian. This provides a lower bound for the class in the general case. We give a simple construction for a set of non-nilpotent metabelian groups which satisfy a finite Engel condition. With Engel class defined by Definition 6·1, we show that there are nilpotent groups of arbitrarily large nilpotency class for which the nilpotency class is equal to the Engel class.

scholarly journals Kp-series and varieties generated by wreath products of p-groups

2018 ◽  
Vol 28 (08) ◽  
pp. 1693-1703
Author(s):  
V. H. Mikaelian
Keyword(s):  
Prime Number ◽  
Wreath Product ◽  
Direct Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Wreath Products ◽  
Product Formula ◽  
Number Formula ◽  
Finite Exponent

Let [Formula: see text] be a nilpotent [Formula: see text]-group of finite exponent and [Formula: see text] be an abelian [Formula: see text]-group of finite exponent for a given prime number [Formula: see text]. Then the wreath product [Formula: see text] generates the variety [Formula: see text] if and only if the group [Formula: see text] contains a subgroup isomorphic to the direct product [Formula: see text] of countably many copies of the cycle [Formula: see text] of order [Formula: see text]. The obtained theorem continues our previous study of cases when [Formula: see text] holds for some other classes of groups [Formula: see text] and [Formula: see text] (abelian groups, finite groups, etc.).

Automorphisms of Iterated Wreath Product p-Groups

2012 ◽  
Vol 55 (2) ◽  
pp. 390-399 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Important Family

AbstractWe determine the order of the automorphism group Aut(W) for each member W of an important family of finite p-groups that may be constructed as iterated regular wreath products of cyclic groups. We use a method based on representation theory.

Burnside metabelian groups

1968 ◽  
Vol 307 (1490) ◽  
pp. 235-250 ◽  
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
General Situation ◽  
Power Exponent ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cyclic Groups ◽  
Number Of Generators

If B is a group of prime-power exponent p e and solubility class 2, then B has nilpotency class at most e ( p e — p e-1 )+1 provided the number of generators of B are at most p +1. Representa­tions of B are constructed which in the case of two generators and prime exponent is a faithful representation of the free group of the variety under study and for prime-power exponent show the existence of a group with nilpotency class e ( p e — p e-1 ). In the general situation where B as above has exponent n , and n is not a prime-power, the place where the lower central series of G becomes stationary is determined by a knowledge of the nilpotency class of the groups of prime-power exponent for all prime divisors of n . The bound e ( p e — p e-1 )+1 on the nilpotency class is a consequence of the following: Let G be a direct product of at most p —1 cyclic groups of order p e and R the group ring of G over the integers modulo p e . Then the e ( p e — p e-1 ) th power of the augmentation ideal of R is contained in the ideal of R generated by all 'cyclotomic’ polynomials Ʃ p e -1 i = 0 g i for all g in G . If G is a direct product of more than p +1 cyclic groups, then this result is no longer true unless e = 1.

scholarly journals BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

2021 ◽  
pp. 1-11
Author(s):  
DMITRY BERDINSKY ◽  
MURRAY ELDER ◽  
JENNIFER TABACK
Keyword(s):  
Computer Science ◽  
Wreath Product ◽  
Cyclic Group ◽  
Wreath Products ◽  
Infinite Cyclic Group ◽  
Cyclic Groups ◽  
Automatic Groups

Abstract We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

scholarly journals Centralisers in Wreath Products

1979 ◽  
Vol 22 (2) ◽  
pp. 161-168 ◽  
Author(s):  
J. D. P. Meldrum
Keyword(s):  
Finite Number ◽  
Wreath Product ◽  
Arbitrary Element ◽  
Wreath Products ◽  
Image Position

In this paper, the centraliser of an arbitrary element of a wreath product is determined. One application of this is to find the breadth of a wreath product (Theorems 21 and 22), a problem which was raised in discussion with Dr. I. D. Macdonald. Another application is to groups generated by elements generating their own centralisers (Theorem 20).Let A and B be two groups. DefineAB = {f : B → A; f(b) = e for all but a finite number of elements of B} to be a group by defining the product pointwise

scholarly journals A characterization of the orders of regressive ω-groups

1972 ◽  
Vol 14 (3) ◽  
pp. 379-382 ◽  
Author(s):  
Matthew J. Hassett
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Cyclic Groups ◽  
Lagrange Theorem ◽  
Finite Subgroups ◽  
Equivalence Type ◽  
Image Position

Let IE, ∧ and ∧R denote the collections of all non-negative integers, isols and regressive isols respectively. An ω-group is a pair (α, p) where (1) α ⊆ E, (2) p(x, y) is a partial recursive group multiplication for α and (3) the function which maps each element of α to its inverse under p has a partial recursive extension. If G = (α, p) is an ω-group, we call the recursive equivalence type of a the RET or order of G (written o(G)). Let GR = {T∈∧R′T = o(G) for some ω-group G}. It follows from the version of the Lagrange Theorem given in [4] that ∧R − GR is non-empty and has cardinality c. In this paper we characterise the isols in GR as follows: A regressive isol T belongs to GR if and only if T∈E or T is infinite and there exist a regressive isol U ≦ T and a function an from E into E − {0} such that U ≦*an and T = ΠUan. (The “≦*” is denned in [2]). In presenting the proof of this result, we shall assume that the reader is familiar with either [3] or [4]. The proof that, given an and U ≦*an, a group of order ΠUan exists is based on the natural trick—one constructs a direct product of disjoint cyclic groups of order a0, a1,… indexed by elements of a set of RET U. The proof that any regressive group G has order of the form ΠUan is trivial for finite groups; the proof for infinite regressive groups is based upon the construction of an ascending chain of finite subgroups Gi of G such that and .

ON GENERATION OF WREATH PRODUCTS OF CYCLIC GROUPS BY TWO STATE TIME VARYING MEALY AUTOMATA

2006 ◽  
Vol 16 (02) ◽  
pp. 397-415 ◽  
Author(s):  
ADAM WORYNA
Keyword(s):  
Topological Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Time Varying ◽  
State Time ◽  
Cyclic Groups ◽  
Mealy Automaton ◽  
Mealy Automata

We proof that an infinitely iterated wreath product of finite cyclic groups of pairwise coprime orders is generated as a topological group by two elements a and b. The group G = 〈a, b〉 may be represented by a 2-state time-varying Mealy automaton. We derive some properties of G.

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